I'm familiar with the mathematics. Cantor's diagonal argument does not make any assertions about the quantity, nature or relationships of infinity. It's one thing to say that there exist two infinite sets for which there is no one-to-one correspondence; it's another beast entirely to claim that one is 'bigger' than the other.

It appears very obvious to me that if it is impossible to count them, then it is false to say that they are countable. — Metaphysician Undercover

It appears very obvious to me that you do not understand the accepted meaning of the word "countable" and, more fundamentally, the distinction between logical possibility and nomological possibility. It is possible in principle to count all of the integers or all of the rational numbers, even though it is not actually possible (as far as we know) for a human being, a machine, or any other physical thing to do so.

No, to say that one is infinitely bigger than the other is nonsense, unless you are assigning spatial magnitude to what is being counted. We are referring to quantities, and each quantity is infinite, how could an infinite quantity be greater than another infinite quantity? — Metaphysician Undercover

No one is talking about spatial magnitude, and talking about numbers does not entail talking about quantities. Your worldview is too small because it limits the real to the actual and the finite.

Why don't you just look it up, or Google it? Plenty of stuff on cardinalities, countable and uncountable infinities, the diagonalization argument, Cantor ... — tom

I have, but you can't believe that just because a mathematician says it is so, therefore it is so. There's a lot of misunderstanding and sophistry in the world.

It appears very obvious to me that you do not understand the accepted meaning of the word "countable" and, more fundamentally, the distinction between logical possibility and nomological possibility — aletheist

As I said, it's an accepted name, "countable". But just because it's called "countable" doesn't means it's actually countable. You seem to believe that it actually does mean that it's countable. And as I explained, when talking specifically about the infinite itself, there is no difference between the countable and the uncountable. There is simply a difference between the thing which you are attempting to count.

Your worldview is too small because it limits the real to the actual and the finite. — aletheist

I'd rather a smaller world view which distinguishes fact from fiction, than a larger world view which doesn't distinguish fact from fiction.

But just because it's called "countable" doesn't means it's actually countable. You seem to believe that it actually does mean that it's countable. — Metaphysician Undercover

Exactly - it actually does mean that it is countable, but it does not mean that it is actually countable. See the difference?

I'd rather a smaller world view which distinguishes fact from fiction, than a larger world view which doesn't distinguish fact from fiction. — Metaphysician Undercover

And I would rather have a worldview that does not make the mistake of treating that which is real as fictional just because it is not actual.

Exactly - it actually does mean that it is countable, but it does not mean that it is actually countable. See the difference? — aletheist

A different word could have been chosen - how about "integer-like" or "zahlen", but that would change nothing. The non-zahlen infinities are vastly bigger, and that is an astronomical understatement.

Exactly - it actually does mean that it is countable, but it does not mean that it is actually countable. See the difference? — aletheist

No I don't see the difference, and you've already tried to explain, but all you do is contradict yourself. "Countable" means possible of being counted. To say that there is a difference between actually countable and potentially countable is nonsense. What would potentially countable mean to you, that it's not countable but could be made to be countable? That's nonsense.

No I don't see the difference, and you've already tried to explain, but all you do is contradict yourself. — Metaphysician Undercover

Show me one genuine contradiction in any of my previous posts, without conflating "countable" (as defined in mathematics) with "actually countable." They are two different concepts.

To say that there is a difference between actually countable and potentially countable is nonsense. — Metaphysician Undercover

To say that there is no difference between actually countable and potentially countable is simply incorrect. Do you really not understand the distinction between the actual and the potential? between the nomologically possible and the logically possible?

Show me one genuine contradiction in any of my previous posts, without conflating "countable" (as defined in mathematics) with "actually countable." They are two different concepts. — aletheist

Try this:

I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. And yet it is a proven mathematical theorem that not only the integers, but also the rational numbers are countable - i.e., it is possible in principle to count them - despite the fact that they are infinitely numerous. — aletheist

See, you say that no one can actually count them, yet it has been proven that it is possible in principle to count them. It's not possible in principle to count them, that's the point, that's what infinite means, that it is impossible to count them. You only contradict yourself.

To say that there is no difference between actually countable and potentially countable is simply incorrect. Do you really not understand the distinction between the actual and the potential? between the nomologically possible and the logically possible? — aletheist

I know very well the difference between potential and actual, as well as many different senses of "possible". It really appears like it's you who has no understanding of this. But if you really believe this is the case, then try to explain the difference between actually countable and potentially countable. Just don't give me contradictions or falsities. If it is impossible to count it, then it is impossible that it is "in principle" countable, because that principle would be a false principle.

See, you say that no one can actually count them, yet it has been proven that it is possible in principle to count them. — Metaphysician Undercover

Yes, and there is no contradiction at all in saying this - unless you insist on conflating "someone can actually count them" with "it is possible in principle to count them," thus refusing to acknowledge that they are NOT the same concept. Counting all of the integers is logically possible, but actually impossible. Infinitely dividing space is logically possible, but actually impossible. Creating a perfect circle is logically possible, but actually impossible. Pure mathematics is the science of drawing necessary conclusions about ideal states of affairs; the actual has nothing to do with it.

You don't seem to understand what "in principle" means. It is impossible to count the infinite, and this is what infinite means, that no matter how you try, you will never ever count it, that's what infinite is. If you now introduce a principle, and say that this principle states that the infinite is countable, such that you can say "it is possible in principle to count them", all you have done is introduced a contradictory principle. It is a false principle

Counting all of the integers is logically possible, but actually impossible. Infinitely dividing space is logically possible, but actually impossible. — aletheist

No. counting all the integers is not logically possible, it is impossible. That's what infinite means, that it is impossible to count them all, you never reach the end. It is such by definition. To say that it is possible to count them all is contradictory. Therefore it is not logically possible.

If you now introduce a principle, and say that this principle states that the infinite is countable, such that you can say "it is possible in principle to count them", all you have done is introduced a contradictory principle. — Metaphysician Undercover

There are really two basic principles here:

• All counting is by means of the natural numbers; therefore, the natural numbers are countable.
• Any set that can be arranged in one-to-one correspondence with the natural numbers, such as the integers, is also countable.

There is nothing contradictory about either of these principles; in fact, together they constitute the very definition of what it means for something to be countable within mathematics. The fact that both the natural numbers and the integers are infinite is completely irrelevant. Think of it this way - it is logically (and actually) impossible to identify a particular integer beyond which it is logically (or actually) impossible to count. If all integers up to any arbitrary finite value are countable, but there is no largest countable integer, then all of the integers must be countable.

No. counting all the integers is not logically possible, it is impossible. — Metaphysician Undercover

One more time: it is logically possible, but actually impossible. You claim to know the difference, but your responses keep indicating otherwise.

That's what infinite means, that it is impossible to count them all, you never reach the end. It is such by definition. — Metaphysician Undercover

That is obviously not what infinite means within mathematics, since the natural numbers and integers are very explicitly defined as countably infinite. You can rail against this terminology all you want, but it will not change the fact that there is no contradiction in saying that the integers are countable as that concept is defined within mathematics.

That is obviously not what infinite means within mathematics, since the natural numbers and integers are very explicitly defined as countably infinite. — aletheist

This is not true. A set is defined as countable if it can be put into bijection with the natural numbers.

By this definition we can then show that the natural numbers, the integers, and the rationals are countable; and that the reals aren't. We define "countable" as a technical term, having no meaning other than that which we've given it. We then prove that the naturals and integers are countable. Formally, having defined the technical term "countable," we then note that the identity map on the naturals, which is a bijection, proves that the natural numbers are countable. Then we prove that the integers are countable by lining them up as 0, 1, -1, 2, -2, ...

As Tom mentioned earlier, much confusion would be avoided if Cantor had picked another name. If we say a set is foozlable if it can be bijected to the natural numbers, then we can prove that the natural numbers, the integers, and the rationals are foozlable; and that the reals aren't. But nobody would have to spend any time arguing about whether you can count the elements of an infinite set.

Surely we all agree that technical terms have specific meanings in context that do not necessarily correspond to their meaning in everyday language. An engineer and a doctor give very different meanings to the word vector. Nobody gets confused, because within their respective technical disciplines the word vector has a formal definition. In the legal profession such words are called "terms of art." A term of art is a word or phrase that has a specific technical meaning within a given discipline that is unrelated to any common meaning.

It's a mistake to think that countability has anything to do with the ability to be counted. That's way too vague. For one thing it's arguably false for the everyday meaning of the word "count." And Cantor's transfinite ordinals let you count way past the natural numbers. Better to simply realize that in set theory, "countable" means exactly one thing and one thing only: that a given set may be bijected to the natural numbers. What Cantor really meant to say is foozlable. Or in the original German, füzlich [That's a joke]. Now any semantic confusion goes away.

As Tom mentioned earlier, much confusion would be avoided if Cantor had picked another name. If we say a set is foozlable if it can be bijected to the natural numbers, then we can prove that the natural numbers, the integers, and the rationals are foozlable; and that the reals aren't. But nobody would have to spend any time arguing about whether you can count the elements of an infinite set. — fishfry

Fair enough, but the fact is that you can count members of a countably infinite set. You can take a subset of a countably infinite set of any number you wish. You can order the set, and you can count from one member to the next.

You can't count the members of an uncountable infinity. There is no such thing as a next member.

Fair enough, but the fact is that you can count members of a countably infinite set. — tom

No, the fact is that you cannot count an infinite set, that's what "infinite" means. You can count a finite subset, but you cannot count the infinite set. "Countable" is just a name, as fishfry explained, it has no other meaning.

You can't count the members of an uncountable infinity. There is no such thing as a next member. — tom

Nor can you count the members of a countable infinity. "Countable" is just the name of the set.

The point I made earlier is that there is actually no difference between the countable infinity and the uncountable, as "infinite", they are the same. What is different is the thing which we are attempting to count, one is a continuity the other discreet units. The continuity cannot be counted, the discrete units can.

Thanks for the excellent clarification/explanation.

A set is defined as countable if it can be put into bijection with the natural numbers. — fishfry

Right, this is all that I meant when I said that the natural numbers are countable by definition. I agree with you that we can then subsequently prove that the integers are also countable.

It's a mistake to think that countability has anything to do with the ability to be counted. — fishfry

Right, this is all that I meant when I said that "countable" is not the same concept as "actually countable." However, I agree with @tom that "you can count members of a countably infinite set"; again, there is no largest natural number or integer beyond which it is (logically or actually) impossible to count, so all of the natural numbers and integers must be countable.

No, the fact is that you cannot count an infinite set, that's what "infinite" means ... The point I made earlier is that there is actually no difference between the countable infinity and the uncountable, as "infinite", they are the same. — Metaphysician Undercover

Incorrect; "uncountable" and "infinite" are not synonyms in mathematics, since there are countable infinities and uncountable infinities. This is a fact, not an opinion.

"Countable" is just a name, as @fishfry explained, it has no other meaning. — Metaphysician Undercover

All words are just names, with no other meanings than how people use and understand them. Mathematicians use and understand "countable" in a very specific way. You do not have to like it, but it is silly to continue insisting otherwise.

What is different is the thing which we are attempting to count, one is a continuity the other discreet units. The continuity cannot be counted, the discrete units can. — Metaphysician Undercover

Not quite, since even the real numbers are still discrete despite being uncountable; they thus form a pseudo-continuum. A true continuum is "that of which every part has parts of the same kind" (Peirce), so it can never be divided into discrete individuals. For example, a truly continuous line can be divided into infinitely many smaller (continuous) lines, but never into (discrete) points.

Yes, this was precisely Bergson's solution. The assumption that space is divisible is a matter of convenience, it does not reflect experience.

Incorrect; "uncountable" and "infinite" are not synonyms in mathematics, since there are countable infinities and uncountable infinities. This is a fact, not an opinion. — aletheist

They are not synonymous, but infinite is by definition not countable. There could be something else uncountable which is not infinite. As we've already discussed, when you refer to countable and uncountable infinities, you use "countable" in a different way, with a different meaning. This way of using "countable" does not imply that a countable infinity is actually countable (according to the other sense of countable), nor does it mean that it is potentially countable, according to the other way of using countable. It is a completely different way of using "countable".

I suggest that you continue to use "countable" in your way, and I'll use "countable" in my way, the two being very obviously incompatible with each other. But you should not claim that you can make the two compatible by saying that one refers to an actuality and the other to a potentiality, because this is not the case. Your sense of "countable infinity" does not equate with "potentially countable" according to my sense of countable, because infinite is neither potentially nor actually countable according to my sense of "countable", it is absolutely uncountable.

However, I agree with tom that "you can count members of a countably infinite set"; again, there is no largest natural number or integer beyond which it is (logically or actually) impossible to count, so all of the natural numbers and integers must be countable. — aletheist

How does this imply that all the natural numbers are countable? It actually implies the very opposite. Every number you count has a larger number, therefore it is impossible that all of the natural numbers are countable. I think you really believe that it is possible to count infinite numbers, because this statement seems to be an attempt to justify this.

I should add that with this simple observation, that space is indivisible, all mathematical theories about nature that rely on mathematical divisibility of space, automatically lose all ontological meaning. They still have practical value but they have no ontological value. This again was Bergson's primary insight into Special and General Relativity.

They are not synonymous, but infinite is by definition not countable. — Metaphysician Undercover

Again, incorrect. You evidently have a rather idiosyncratic personal definition of "infinite." My dictionary provides several widely accepted definitions, and none of them state or imply that it means "not countable." Besides, as I keep noting, the concept of being "countably infinite" is well-established and well-understood within mathematics.

But you should not claim that you can make the two compatible by saying that one refers to an actuality and the other to a potentiality, because this is not the case. — Metaphysician Undercover

I have never claimed that our different definitions of "countable" are compatible. I have simply demonstrated that my definition is not contradictory, and that yours is simply wrong, at least within mathematics. Whether something is actually possible is completely irrelevant when dealing with ideal states of affairs, which is all that pure mathematics ever does.

How does this imply that all the natural numbers are countable? — Metaphysician Undercover

Because you can always keep counting beyond any arbitrary finite value; i.e., you cannot identify a single natural number or integer that is uncountable. Obviously, if there are no uncountable natural numbers or integers, then all of the natural numbers and integers are countable.

Every number you count has a larger number, therefore it is impossible that all of the natural numbers are countable. — Metaphysician Undercover

Now you seem to be confusing "countable" with the idea of being finished counting. This is not what "countable" means within mathematics, either. That larger number is just as countable as the one that you already counted; and so is the next larger number; and so on, ad infinitum - which is the whole point.

I think you really believe that it is possible to count infinite numbers, because this statement seems to be an attempt to justify this. — Metaphysician Undercover

I have stated plainly (and repeatedly) that I believe this to be logically possible, but not actually possible.

The assumption that space is divisible is a matter of convenience, it does not reflect experience. — Rich

I should add that with this simple observation, that space is indivisible, all mathematical theories about nature that rely on mathematical divisibility of space, automatically lose all ontological meaning. — Rich

What exactly do you mean when you assert that "space is indivisible"? Are you merely saying (as I do) that space is continuous, rather than discrete - i.e., it cannot be divided into dimensionless points, only smaller and smaller three-dimensional spaces? Or do you have something else in mind?

What I mean to say is that carving up space, in imagination, is a matter of convenience, mathematical or otherwise. Space is continuous as is real time (duration). Mathematics is simply a symbolic tool to aid in measurement (along with actual observations). However, scientific equations themselves have no ontological value simply because of this disconnect with actual experience. Using mathematical equations as a substitute for actual experience leads to all kinds of paradoxes such as time travel, twins aging differently, and of course Zeno's.

Bohm pointed out that where there are paradoxes there are some really nasty problems with assumptions.

It sounds like we are on the same page here. As Charles Sanders Peirce put it, citing his father:

I do not know that anybody struck the true note before Benjamin Peirce, who, in 1870, declared mathematics to be 'the science which draws necessary conclusions' ... the essence of mathematics lies in its making pure hypotheses, and in the character of the pure hypotheses which it makes. What the mathematicians mean by a 'hypothesis' is a proposition imagined to be strictly true of an ideal state of things. In this sense, it is only about hypotheses that necessary reasoning has any application; for, in regard to the real world, we have no right to presume that any given intelligible proposition is true in absolute strictness.

You can't count the members of an uncountable infinity. There is no such thing as a next member. — tom

You can certainly well-order an uncountable set. You need the Axiom of Choice to well-order the real numbers, but you do not need Choice to show the existence of the first uncountable ordinal. That Wiki page is light on detail but the idea is that the set of all countable ordinals is an ordinal (needs proof of course), and it can't be a countable one (because a set can't be a member of itself), hence it must be an uncountable ordinal. Such a thing is impossible to visualize but it exists.

An ordinal is an order type of a well-ordered set. A set is well-ordered if every nonempty subset has a smallest member. There's a first, then a second, then a third, etc. Clearly the natural numbers are well-ordered. Now to get to larger ordinals you have to allow limit ordinals, which are unions of upward chains of ordinals. I don't want to get technical, which is why in my earlier post I just wrote

And Cantor's transfinite ordinals let you count way past the natural numbers. — fishfry

That's why the usage of the everyday meaning of counting is totally out of place here. It's vague, and mathematicians can indeed well-order uncountable sets.

Right, this is all that I meant when I said that the natural numbers are countable by definition. — aletheist

No I'm afraid you are still missing my point. I defined a set as countable if it can be put into bijection with the naturals. You claim this "defines" the naturals as countable but I say, "I don't believe you. Prove it." And you say: "Aha, the identity function on the naturals is a bijection." You have PROVED directly from the definition that the naturals are countable. It's a theorem (admittedly so easy it's never stated explicitly) and not a definition.

No I'm afraid you are still missing my point. — fishfry

No, I get it, you just stated more accurately what I meant all along. :)

There are areas of the quote that I would agree with and there are others that I would phrase differently. Utmost is the issue that science way over steps it's bounds when it begins to replace everyday experiences with symbolic equations and declaring the equations to be more real. If such was true, we couldn't move or more ernestly we all become illusions of equations. This is what Bergson and Bohm objected to.

Utmost is the issue that science way over steps it's bounds when it begins to replace everyday experiences with symbolic equations and declaring the equations to be more real. — Rich

Yes, we must always keep in mind that such equations are models - or in Peirce's terminology, diagrams - which embody only the parts and relations within the actual situation that someone has deemed to be significant. Consequently, they are only as "accurate" as this judgment on the part of the modeler and the underlying assumptions of the selected representational system, including its transformation rules.

Yes, this I would fully agree with.

I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. And yet it is a proven mathematical theorem that not only the integers, but also the rational numbers are countable - i.e., it is possible in principle to count them - despite the fact that they are infinitely numerous. — aletheist

I went back through this thread from the beginning. Finally on page 11, this quote is the first mention of mathematical countability. The above quote is simply flat out wrong. It commits the fallacy (does it have a formal name?) of confusing a term of art with its everyday meaning. Countability as defined in mathematics simply has nothing at all to do with the everyday meaning of the ability to be counted. I already made this point but now I found the source of the recent confusion in this thread.

A child learning to count, "one, two three, four, ..." has absolutely nothing to do with mathematical countability. Saying that a set is countable does NOT mean "it is possible in principle to count them." It means exactly that there exists a bijection from the natural numbers to the set. Nothing more and nothing less.

You know the old joke. "Why can't you cross a mountain climber with a mosquito? Because you can't cross a scaler with a vector." That joke depends on conflating the engineering definitions of scalar, vector and cross (as in cross product) with the common English meaning of a climber -- a "scaler" -- and the medical meaning of vector -- a means of disease transmission, and the biological meaning of cross, as to cross-breed living things based on their genetic makeup.

But this is a JOKE, not something you can take seriously in a philosophical discussion. You can not, unless you being disingenuous, say that "The rational numbers are countable" and then say this shows that a child could count them in the every day sense of the word.

If you counted, in the sense of saying out loud "one, two, three ..." the natural numbers, starting at the moment of the Big Bang, at the rate of a number per second; or ten numbers, or a trillion -- you would not finish before the heat death of the universe.

You are simply conflating a term of art -- a technical term used with a specific meaning in a specific context by specialists -- with the everyday meaning of the term.

Sorry to be ranting now but really, the quote above is terribly wrong. You can't count the natural numbers in the every day meaning of the word. There are infinitely many of them. The natural numbers are countable, in the technical sense that there exists a bijection between the natural numbers and themselves. If you think to yourself, "The natural numbers, the integers, and the rational numbers are examples of foozlable sets," you will not confuse yourself or others by shifting the meaning of a technical term to its everyday meaning.